A stable numerical method for the dynamics of fluidic membranes

We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier–Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier–Stokes equation, taking surface viscosity effects of Boussinesq–Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (2008). The obtained numerical scheme can be shown to be stable and to have good mesh properties. Finally, the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions. The numerical results demonstrate the robustness of the method, and it is observed that the conservation properties are fulfilled to a high precision

A stable numerical method for the dynamics of fluidic membranes

We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier–Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier–Stokes equation, taking surface viscosity effects of Boussinesq–Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (2008). The obtained numerical scheme can be shown to be stable and to have good mesh properties. Finally, the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions. The numerical results demonstrate the robustness of the method, and it is observed that the conservation properties are fulfilled to a high precision

Solving the incompressible surface Navier-Stokes equation by surface finite elements

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincar\'e-Hopf theorem on $n$-tori

Accurate Multi-physics Numerical Analysis of Particle Preconcentration Based on Ion Concentration Polarization

This paper studies mechanism of preconcentration of charged particles in a straight micro-channel embedded with permselective membranes, by numerically solving coupled transport equations of ions, charged particles and solvent fluid without any simplifying assumptions. It is demonstrated that trapping and preconcentration of charged particles are determined by the interplay between drag force from the electroosmotic fluid flow and the electrophoretic force applied trough the electric field. Several insightful characteristics are revealed, including the diverse dynamics of co-ions and counter ions, replacement of co-ions by focused particles, lowered ion concentrations in particle enriched zone, and enhanced electroosmotic pumping effect etc. Conditions for particles that may be concentrated are identified in terms of charges, sizes and electrophoretic mobilities of particles and co-ions. Dependences of enrichment factor on cross-membrane voltage, initial particle concentration and buffer ion concentrations are analyzed and the underlying reasons are elaborated. Finally, post priori a condition for validity of decoupled simulation model is given based on charges carried by focused charge particles and that by buffer co-ions. These results provide important guidance in the design and optimization of nanofluidic preconcentration and other related devices.Comment: 18 pages, 11 firgure

Microfluidic systems for in situ formation of nylon 6,6 membranes.

A microfluidics based, localised formation of nylon 6,6 membranes has been undertaken. The study demonstrates the feasibility of maintaining stable aqueous/organic interfaces for xylene within simple linear flow channels. Glass fabricated structures were used with adipoyl chloride and hexamethylenediamine in the organic and aqueous phases, respectively, in order to achieve nylon 6,6 interfacial polymerisation. Localised membrane formation was investigated in flow channels of different geometries over a wide range of flow rates (500–4000 μl/min), with Reynolds numbers ranging from 8.4 to 67.2. The results demonstrate that interfacial polymerisation occurs consistently over a wide range of flow rates and of flow entry angles for dual aqueous/organic solvent input. However, creation of uniform planar film structures required careful optimisation, and these were best achieved at 2000 μl/min with a flow entry angle of 45°. The resulting membranes had thicknesses in the range between 100 and 300 μm. Computational modelling of the aqueous/organic flow was performed in order to characterise flow stability and wall shear-stress patterns. The flow arrangement establishes a principle for the fabrication of micromembrane structures designed for low sample volume separation, where the forming reaction is a facile and rapid interfacial process

A Trace Finite Element Method for Vector-Laplacians on Surfaces

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface-independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included

Computing stationary free-surface shapes in microfluidics

A finite-element algorithm for computing free-surface flows driven by arbitrary body forces is presented. The algorithm is primarily designed for the microfluidic parameter range where (i) the Reynolds number is small and (ii) force-driven pressure and flow fields compete with the surface tension for the shape of a stationary free surface. The free surface shape is represented by the boundaries of finite elements that move according to the stress applied by the adjacent fluid. Additionally, the surface tends to minimize its free energy and by that adapts its curvature to balance the normal stress at the surface. The numerical approach consists of the iteration of two alternating steps: The solution of a fluidic problem in a prescribed domain with slip boundary conditions at the free surface and a consecutive update of the domain driven by the previously determined pressure and velocity fields. ...Comment: Revised versio